Area of a triangle is the region enclosed by it, in a two-dimensional plane. As we know, a triangle is a closed shape that has three sides and three vertices. Thus, the area of a triangle is the total space occupied within the three sides of a triangle. The general formula to find the area of the triangle is given by half of the product of its base and height.
What is Area of a Triangle?
The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h. Hence, to find the area of a tri-sided polygon, we have to know the base (b) and height (h) of it. It is applicable to all types of triangles, whether it is scalene, isosceles or equilateral. To be noted, the base and height of the triangle are perpendicular to each other. The unit of area is measured in square units (m2, cm2).
Example: What is the area of a triangle with base b = 3 cm and height h = 4 cm?
Using the formula,
Area of a Triangle, A = 1/2 × b × h = 1/2 × 4 cm × 3 cm = 2 cm × 3 cm = 6 cm2
Apart from the above formula, we have Heron’s formula to calculate the triangle’s area, when we know the length of its three sides. Also, trigonometric functions are used to find the area when we know two sides and the angle formed between them in a triangle. We will calculate the area for all the conditions given here.
Area of a Triangle Formula
The area of the triangle is given by the formula mentioned below:
Area of a Triangle = A = ½ (b × h) square units
Area of a Right Angled Triangle
A right-angled triangle, also called a right triangle has one angle at 90° and the other two acute angles sums to 90°.
Area of a Right Triangle = A = ½ × Base × Height (Perpendicular distance)
From the above figure,
Area of triangle ACB = 1/2 ab
Area of an Equilateral Triangle
An equilateral triangle is a triangle where all the sides are equal. The perpendicular drawn from the vertex of the triangle to the base divides the base into two equal parts
- Area of an Equilateral Triangle = A = (√3)/4 × side2
Area of an Isosceles Triangle
An isosceles triangle has two of its sides equal and also the angles opposite the equal sides are equal.
- Area of an Isosceles Triangle = 1/4 b√(4a2 – b2)
Perimeter of a Triangle
- The perimeter of a triangle = P = (a + b + c) units
Area of Triangle with Three Sides (Heron’s Formula)
The area of a triangle with 3 sides of different measures can be found using Heron’s formula. Heron’s formula includes two important steps.
where, s is semi-perimeter of the triangle = s = (a+b+c) / 2
Area of a Triangle Given Two Sides and the Included Angle (SAS)
Now, the question comes, when we know the two sides of a triangle and an angle included between them, then how to find its area.
Now, if any two sides and the angle between them are given, then the formulas to calculate the area of a triangle is given by:
Area (∆ABC) = ½ bc sin A
Area (∆ABC) = ½ ab sin C
Area (∆ABC) = ½ ca sin B
These formulas are very easy to remember and also to calculate.
For example, If, in ∆ABC, A = 30° and b = 2, c = 4 in units. Then the area will be;
Area (∆ABC) = ½ bc sin A
= ½ (2) (4) sin 30
= 4 x ½ (since sin 30 = ½)
= 2 sq.unit.
Area of a Triangle Solved Examples
Example 1:
Find the area of an acute triangle with a base of 13 inches and a height of 5 inches.
Solution:
A = (½)× b × h sq.units
⇒ A = (½) × (13 in) × (5 in)
⇒ A = (½) × (65 in2)
⇒ A = 32.5 in2
